An LC circuit contains a 20 mH inductor and a `50 mu F` capacitor with an initial charge of 10 mC. The resistance of the circuit is negligible. Let the instant the circuit is closed be t = 0.
(a) What is the total energy stored initially ? Is it conserved during the oscillalions?
(b) What is the natural frequency of the circuit?
(c ) At what time is the energy stored? (i) Completely electrical ? (ii) Completely magnetic ?
(d) At what time is the total energy shared equally between the inductor and the capacitor ?
(e) If a resistor is inserted in the circuit, how much energy is eventually dissipated as heat ?

Explanation

(a) 1.0 J. Yes, sum of the energies stored in L and C is conserved if R = 0.
(b) `omega=10^(3)rad s^(-1), v=159Hz`
(c ) `q=q_(0)cos omega t`
(i) Energy stored is completely electrical at `t=0, (T)/(2), T, (3T)/(2), ……`
(ii) Energy stored is completely magnetic (i.e., electrical energy is zero) at `t=(T)/(4), (3T)/(4), (5T)/(4).......`, where `T=(1)/(v)=6.3ms`.
(d) At `(T)/(8), (3T)/(8), (5T)/(8) ..........`, because `q=q_(0)"cos"(omegaT)/(8) =q_(0)"cos"(pi)/(4)=(q_(0))/(sqrt(2))`.
Therefore, electrical energy `=(q^(2))/(2C)=(1)/(2)((q_(0)^(2))/(2C))` which is half the total energy.
R damps out the LC oscillations eventually. The whole of the initial energy (= 1.0 J) is eventually dissipated as heat.